A proof with sums

1. Mar 25, 2010

1. The problem statement, all variables and given/known data
Let {a_n} be an alternating series of real numbers approaching zero. Prove there exist a z0 on the unit circle such that the power series $$\sum_{}^{} a_{n}z^{n}$$ is uniformly convergent on the domain of z such that both |z|<=1 and |z-z0|>= delta, where delta>0

2. Relevant equations

no idea, especially the thing about "Prove that such a z0 exist": No idea what I can do with that..

3. The attempt at a solution
For those who have Serge Langs complex analysis: See page 454 appendix I, propostion 1.2b, which was given as a hint, but I don't have a clue from where I can show such a z0 exist ..?
I thought that perhaps you can do it by contradiction: But what would be the negation..?
Sorry, but I am very confused, help, anyone?